# Full text of "Treatise On Analysis Vol-Ii"

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```224       XIII   INTEGRATION

inequality in the reverse direction (with equality when h e &*(X x Y)) for
lower integrals:

(13.21.4.3)      IT /z(x, y) dl(x) dn(y) g f dl(x) !h(x, y) dfi(y).

(13.21.5) 7/*N is a v-negligible subset ofXx Y, then the set of points x e X
such that the section N(^) c: Y of N is not ^-negligible is ^-negligible (in
other words, we have ju(N(.x)) = 0 almost everywhere with respect to A).

This follows immediately from (13.21.4.2) applied to h = <pN.

(13.21 .6) If his a ^-measurable mapping ofX x Y into a topological space E,
then the set of points xeX for which the partial mapping yt-*h(x, y) is not
^-measurable is ^-negligible (in other words, the mapping y\-*h(x9y) is ju-
measurable almost everywhere with respect to A).

By hypothesis, there exists a partition of X x Y consisting of a v-negligible
set N and a sequence (Kn)B^i of compact sets, such that each of the restric-
tions A|KB is continuous (13.9). Let M be the A-negligible set of points
;ceX at which the section N(x) of N is not ju-negligible (13.21.5). For each
x \$ M, Y admits a partition consisting of compact sets Kn(x) (n ^ 1) and
the /^-negligible set N(x), such that the restriction of y\->h(x, y) to each
of the sets KB(#) is continuous. Hence the result.

It should be noted that it can happen that for each x e X the function
y*-*f(x,y) is ^-measurable, and for each y e Y the function x\-+f(x,y) is
A-measurable, but that f is not v-measurable.

(1 3.21 .7) (Lebesgue-Fubini Theorem) Let 1, \i be positive measures on X
and Y, respectively, and v = A ® \i their product. For each \-integrable map-
ping h o/X x Y into R, the set of points x e X such that the partial mapping

yt-*h(x,y) is not ^-integrabte is ^-negligible; the function XH-> \h(x,

o                                                                    '                        J

which is defined almost everywhere with respect to A, is X-integrable ; and

(12.21.7.1)           h(x, y) dl(x

It follows from (13.21.4.1) that the function xi~» f*|A(#, y)\ dfj,(y) is finite
on the complement of a A-negligible set Nx (13.6.4). On the other hand, the
set N2 of points x e X such that y H> h(x, y) is not ju-measurable is A-negligibleion of the (finitely many) interiors of sets VjO^jCj)) which contain y; the
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